Let $\theta$ be the angle between the vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ where $|\overrightarrow{\mathrm{a}}|=4,|\overrightarrow{\mathrm{~b}}|=3 \theta \in\left(\frac{\pi}{4}, \frac{\pi}{3}\right)$. Then

If the shortest distance between the line $\vec{r}=(-\hat{i}+3 \hat{k})+\lambda(\hat{i}-a \hat{j})$ and $\vec{r}=(-\hat{j}+2 \hat{k})+\mu(\hat{i}-\hat{j}+\hat{k})$ is $\sqrt{\frac{2}{3}}$ then the integral value of is equal to

Let $Q$ be the mirror image of the point $P(1,0,1)$ with respect to the plane $S: x+y+z=5$. If a line $L$ passing through $(1,-1,-1)$, parallel...

Let a line having direction ratios $1,-4,2$ intersect the lines $\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}$ and $\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}$ at the point $A$ and $B$. Then $(A B)^2$ is equal to...

Let $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k} \quad a_i>0, i=1,2,3$ be a vector which makes equal angles with the coordinates axes OX, OY and OZ . Also,...

Let $\hat{a}, \hat{b}$ be unit vectors. If $\vec{c}$ be a vector such that the angle between $\hat{a}$ and $\hat{c}$ is $\frac{\pi}{12}$, and $\hat{b}=\vec{c}+2(\vec{c} \times \hat{a})$,...

Let $P_1$ be the plane $3 x-y-7 z=11$ and $P_2$ be the plane passing through the points $(2,-1,0),(2,0,-1)$, and $(5,1,1)$. If the foot of the...

Let the vectors $\overrightarrow{\mathrm{u}}_1=\hat{\mathrm{i}}+\hat{\mathrm{j}}+a \hat{\mathrm{k}}, \overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+b \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{u}}_3=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be coplanar. If the vectors $\vec{v}_1=(a+b) \hat{i}+c \hat{j}+c \hat{k}, \quad \vec{v}_2=a \hat{i}+(b+c) \hat{j}+a \hat{k}$ and $...

Let the area of the triangle formed by the lines $x+2=y-1=z, \frac{x-3}{5}=\frac{y}{-1}=\frac{z-1}{1}$ and $\frac{x}{-3}=\frac{y-3}{3}=\frac{z-2}{1}$ be $A$. Then $A^2$ is equal to $\_\_\_\_$ .

For $\mathrm{a}, \mathrm{b} \in \mathrm{Z}$ and $|\mathrm{a}-\mathrm{b}| \leq 10$, let the angle between the plane $\mathrm{P}: \mathrm{ax}+\mathrm{y}-\mathrm{z}=\mathrm{b}$ and the line $l: \mathrm{x}-1=\mathrm{a} -y=z+1$ be $...